## Andrew Ranicki

Print publication date: 2002

Print ISBN-13: 9780198509240

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198509240.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use.  Subscriber: null; date: 23 October 2019

# THE ODD-DIMENSIONAL SURGERY OBSTRUCTION

Chapter:
(p.302) 12 THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
Source:
Algebraic and Geometric Surgery
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509240.003.0012

# Abstract and Keywords

This chapter provides the algebraic construction and geometric properties of the odd-dimensional surgery obstruction groups. It includes quadratic and kernel forms.

This chapter defines the Wall surgery obstruction of a (2n+1)-dimensional degree 1 map (f, b) : MX,

$Display mathematics$
The main result is that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is normal bordant to a homotopy equivalence.

Section 12.1 introduces the notion of an ε-quadratic formation (K, λ, μ; F, G), which is a nonsingular ε-quadratic form (K, λ, μ) with lagrangians F, G. Section 12.2 constructs a kernel (−1)n-quadratic formation over Z[π1(X)] for an n-connected (2n + 1)-dimensional degree 1 normal map (f, b). The (2n + 1)-dimensional surgery obstruction group L 2n+1(A) of cobordism classes of nonsingular (−1)n-quadratic formations (K, λ, μ; F, G) over A is defined in Section 12.3. The odd-dimensional surgery obstruction is defined in Section 12.4 to be the cobordism class of a kernel formation. Section 12.5 describes the algebraic effect on a kernel formation of a geometric surgery on (f, b). Finally, Section 12.6 gives a brief account of linking forms, the original approach to odd-dimensional surgery obstruction theory.

Odd-dimensional surgery obstruction theory is technically more complicated than the even-dimensional case. Specifically, an n-connected 2n-dimensional normal map has a unique kernel form, whereas an n-connected (2n + 1)-dimensional normal map has many kernel formations.

The surgery obstruction of an even-dimensional degree 1 normal map was expressed in Section 11.5 as the equivalence class of the kernel nonsingular ε-quadratic form, with the zero class represented by the forms which admit a lagrangian. The different lagrangians admitted by the kernel form correspond to different ways of solving the even-dimensional surgery problem by a normal bordism to a homotopy equivalence. The odd-dimensional surgery obstruction will be expressed in Section 12.4 as an equivalence class of ε-quadratic ‘formations’, which are nonsingular ε-quadratic forms with ordered pairs of lagrangians, corresponding to two solutions of an even-dimensional surgery problem in codimension 1.

(p.303) The fundamental algebraic structure determined by a closed (n − 1)-connected 2n-dimensional manifold N 2n is the nonsingular (−1)n-symmetric form (H n(N),λ) over Z. The fundamental algebraic structure determined by an (n − 1)-connected (2n + 1)-dimensional manifold with boundary (M 2n+1,∂M) with H n(M, ∂M) = 0 is the lagrangian of the (−1)n-symmetric form (H n(∂M),λ) defined by

$Display mathematics$
Now suppose that M is a closed (2n+1)-dimensional manifold which is expressed as a union of two (2n+1)-dimensional manifolds $( M + 2 n + 1 , ∂ M + ) , ( M − 2 n + 1 , ∂ M − )$ with the same (n − 1)-connected boundary N 2n = ∂M + = ∂M -:
If M, M +, M -, N are (n − 1)-connected and H n(M +, N) = H n(M -, N) = 0 the (−1)n-symmetric form on K = H n(N) has lagrangians
$Display mathematics$
giving a(-1)n - symmetric formation (K,λ;L+L-). Such splitings were first used by Heegaard (in 1898) in the study of 3-diamensional manifolds: every conected Mainfold M3 can be expressed as a union,
$Display mathematics$
of solid tori along a genus g surface, so that
$Display mathematics$
corresponding to a handle decomposition
$Display mathematics$
(p.304) It should be clear that such expressions are not unique, since forming the connected sum of M with
$Display mathematics$
increases the genus g by 1 without affecting the diffeomorphism type.

The remainder of this section will only consider the algebraic properties of formations. As before, let A be a ring with involution, and let ε = ±1.

Definition 12.1 An ε-quadratic formation over A (K, λ, μ; F, G) is a nonsingular ε-quadratic form (K, λ, μ) together with an ordered pair of lagrangians F, G. □

Strictly speaking, Definition 12.1 defines a ‘nonsingular formation’. In the general theory a formation (K, λ, μ; F, G) is a nonsingular form (K, λ, μ) together with a lagrangian F and a sublagrangian G, with F, G, and K f.g. projective. For basic odd-dimensional surgery obstruction theory only nonsingular formations with F, G, and K f.g. free need be considered. Also, A can be assumed to be such that the rank of f.g. free A-modules is well-defined (e.g. A = Z[π]), with A k isomorphic to A l if and only if k = l. This hypothesis ensures that for every formation (K, λ, μ; F, G) over A there exists an automorphism a: (K, λ, μ) → (K, λ, μ) such that α(F) = G (Proposition 12.3 below). In the original work of Wall [92, Chapter 6] the odd-dimensional surgery obstruction was defined in terms of such automorphisms. Novikov [64] proposed the use of pairs of lagrangians instead, and the expression of odd-dimensional surgery obstructions in terms of formations was worked out in Ranicki [68].

Definition 12.2 An isomorphism of ε-quadratic formations over A,

$Display mathematics$
is an isomorphism of forms f: (K, λ, μ) → (K′, λ′, μ′) such that
$Display mathematics$
□

Proposition 12.3

1. (i) Every ε-quadratic formation (K, λ, μ; F, G) is isomorphic to one of the type (H ε(F); F, G).

2. (ii) Every ∈-quadratic formation (K, λ, μ; F, G) is isomorphic to one of the type (H (F); F, α(F)) for some automorphism α: H ε(F) → H ε(F).

Proof (i) By Theorem 11.51 the inclusion of the lagrangian FK extends to an isomorphism of ε-quadratic forms,

$Display mathematics$
defining an isomorphism of ε-quadratic formations,
$Display mathematics$

(p.305) (ii) As in (i) extend the inclusions of the lagrangians F → K, G → K to isomorphisms of forms

$Display mathematics$
Then,
$Display mathematics$
so that F is isomorphic to G. Choosing an A-module isomorphism β: F → G there is defined an automorphism of H (F)
$Display mathematics$
such that there is defined an isomorphism of formations
$Display mathematics$
□

In Section 12.2 there is associated to an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X a ‘stable isomorphism’ class of kernel (−1)n-quadratic formations (K, λ, μ; F, G) over Z[π1(X)] such that

$Display mathematics$
Stable isomorphism is defined as follows:

Definition 12.4

1. (i) An ε-quadratic formation T = (K, λ, μ; F, G) is trivial if it is isomorphic to (H ε(F); F, F*).

2. (ii) A stable isomorphism of ε-quadratic formations,

$Display mathematics$
is an isomorphism of ε-quadratic formations of the type
$Display mathematics$
with T, T′ trivial. □

Proposition 12.5

1. (i) An ε-quadratic formation (K, λ, μ; F, G) is trivial if and only if the lagrangians F and G are direct complements in K:

$Display mathematics$

2. (p.306)
3. (ii) A stable isomorphism [f]: (K, λ, μ; F, G) → (K′, λ′, μ′; F′, G′) of ε-quadratic formations over A induces A-module isomorphisms

$Display mathematics$

4. (iii) For any ε-quadratic formation (K, λ, μ; F, G) there is defined a stable isomorphism

$Display mathematics$

Proof (i) If F and G are direct complements in K, represent (λ, μ) by a split ε-quadratic form (K, ψ ∈ S(K)) with

$Display mathematics$
Then, γ + εδ* ∈ HomA(G, F *) is an A-module isomorphism, and there is defined an isomorphism of formations
$Display mathematics$

(ii) By (i) an ε-quadratic formation (K, λ, μ; F, G) is trivial if and only if

$Display mathematics$

(iii) By Proposition 12.3 we can take

$Display mathematics$
with
$Display mathematics$
an isomorphism of hyperbolic ε-quadratic forms, which defines an isomorphism of ε-quadratic formations
$Display mathematics$
The A-module isomorphism,
$Display mathematics$
defines an isomorphism of ε-quadratic formations,
$Display mathematics$
giving a stable isomorphism
$Display mathematics$
□

# (p.307) 12.2 The kernel formation

An n-connected (2n+1)-dimensional degree 1 normal map (f, b): M 2n+1X is such that K i(M) = 0 for in, n + 1. We shall now construct a stable isomorphism class of kernel (−1)n-quadratic formations (K, λ, μ; F, G) over Z[π1(X)] such that

$Display mathematics$
using the following generalization of the Heegaard splitting of a 3-dimensional manifold as a union of solid tori.

The kernel Z[π1 (X)]-module K n (M) is finitely generated, by Corollary 10.29. Every finite set {x 1, x 2, …,x k} of generators is realized by disjoint n-surgeries

for i = 1, 2,…, k.

By the Poincaré Disc Theorem (9.14) it may be assumed that the target (2n+ 1)-dimensional geometric Poincare complex X is obtained from a (2n + 1)- dimensional geometric Poincaré pair (X 0, S 2n) by attaching a (2n + 1)-cell

$Display mathematics$
Any choice of generators x 1, x 2, …, xkKn(M) is realized by framed embeddings gi: Sn × D n+1M 2n+1 together with null-homotopies hi: fgi ≃ {*}: SnX.

Definition 12.6 A Heegaard splitting of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is an expression as a union

$Display mathematics$
with
$Display mathematics$
associated to a set of framed n-embeddings
$Display mathematics$
with null-homotopies in X representing a set {x 1, x 2, …, xk} ⊂ Kn(M) of Z[π1(X)]-module generators. □ (p.308)

Every finite set of generators of K n(M) is realized by a Heegaard splitting.

Definition 12.7 The kernel formation of an n-connected (2n+1)-dimensional degree 1 normal map (f, b): M 2n+1X with respect to a Heegaard splitting is the (−1)n-quadratic formation over Z[π1(X)],

$Display mathematics$
with $( K n ( ∂ U ) , λ , μ ) = H ( − 1 ) n ( K n + 1 ( U , ∂ U ) )$ the hyperbolic (–1)n-quadratic kernel form over Z[π1(X)] of the n-connected 2n-dimensional degree 1 normal map ∂US 2n. The lagrangians F, G are determined by the n-connected (2n + 1)-dimensional degree 1 normal maps of pairs
$Display mathematics$
with
$Display mathematics$
□

The Heegaard splittings and kernel formations of an n-connected (2n + 1)- dimensional degree 1 normal map (f, b): M 2n+1X are highly non-unique.

Proposition 12.8 For any kernel formation (K, λ, μ; F, G) of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X there are natural identifications of Z[π1(X)]-modules:

$Display mathematics$

Proof Immediate from the exact sequence

$Display mathematics$
and the identification of the map in the middle with the natural Z[π1(X)]-module morphism
$Display mathematics$
□

(p.309) The kernel formations will be used in Section 12.4 to represent the surgery obstruction σ *(f, b) = (K, λ, μ; F, G) ∈ L 2n+1(Z[π1(X)]).

Example 12.9 The kernel formation of the 1-connected degree 1 normal map on the 3-dimensional lens space (8.39),

$Display mathematics$
is the (−1)-quadratic formation over Z given by
$Display mathematics$
In particular,
1. (i) L(1, 0) = S 3 with trivial formation $( H − ( ℤ ) ; ℤ , ( 0 1 ) ℤ ) ,$

2. (ii) L(0, 1) = S 1 × S 2 with boundary formation $( H − ( ℤ ) ; ℤ , ( 1 0 ) ℤ ) ,$

3. (iii) L(1, 2) = S 3 with trivial formation $( H − ( ℤ ) ; ℤ , ( 2 1 ) ℤ ) ,$

4. (iv) L(2, 1) = R P 3 with boundary formation $( H − ( ℤ ) ; ℤ , ( 1 2 ) ℤ ) .$ . □

Proposition 12.10 Let (f, b): M 2n+1X be an n-connected (2n+1)-dimensional normal map.

1. (i) The kernel formations associated to all the Heegaard splittings of (f, b) are stably isomorphic.

2. (ii) For n ⩾ 2 every formation in the stable isomorphism class is realised by a Heegaard splitting of (f, b).

3. (iii) A kernel formation of (f, b) is trivial if (and for n ⩾ 2 only if) (f, b) is a homotopy equivalence. □

Proof (i) Consider first the effect on the kernel formation of changing the framed n-embeddings g i: S n × D n+1M 2n+1 representing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators for K n (M). Proposition 10.13 gives

$Display mathematics$
so that the framed n-embeddings g i are unique up to regular homotopy. Any two sets of framed n-embeddings representing x i,
$Display mathematics$
are thus related by regular homotopies
$Display mathematics$
(p.310) with null-homotopies in X. Write the inclusions of the lagrangians associated to the two Heegaard splittings
$Display mathematics$
as
$Display mathematics$
Let
$Display mathematics$
be the Z[π1(X)]-module isomorphisms determined by the given basis elements, and let (F*, ψ) be a kernel split (−)n+1-quadratic form for the immersion
$Display mathematics$
(defined as in the proof of Proposition 11.42). The commutative diagram
defines an isomorphism of the kernel (−1)n-quadratic formations
$Display mathematics$

Next, consider the relationship between the kernel formations associated to two different sets of generators of K n(M). Proceed as in Wall [92,Chapter 6]. (p.311) Any two sets {x 1, x 2, …, x k}, {y 1, y 2, …, y l} of Z[π1(X)]-module generators for K n(M) are related by a sequence of elementary operations

$Display mathematics$
with $y i = ∑ j a i j x j$ for some a ij ∈ Z[π1(X)]. Each of these operations has one of the following types:
1. (1) adjoin (or delete) a zero,

2. (2) permute the elements,

3. (3) add to the last element a Z[π1(X)]-linear combination of the others.

The effect of (1) on the kernel formation of (f, b) is to add (or delete) a trivial formation
$Display mathematics$
while (2) and (3) do not change it.

(ii) For n ⩾ 2 it is possible to realize every elementary operation geometrically.

(iii) A kernel formation for (f, b) is trivial if and only if K *(M) = 0. Now K *(M) = 0 if (and for n ⩾ 2 only if) (f, b) is a homotopy equivalence. □

The kernel formations of (f, b): M 2n+1X were obtained in 12.7 by working inside M, using Heegaard splittings. However, as described in Example 6.3 of Ranicki [75] (and pp. 71–72 of the second edition of Wall [92]) there is an alternative construction, working outside of M as follows.

Definition 12.11 A presentation of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is a degree 1 normal bordism

$Display mathematics$
such that f′: M′ → X × {1} is n-connected and e: W → X × I is (n+1)-connected. □

The (2n + 2)-dimensional manifold with boundary W in a presentation has a handle decomposition on M of the type

$Display mathematics$
(p.312) The kernels K n+1(W), K n+1(W, M), K n+1(W, M′) are f.g. free Z[π1(X)]-modules of rank k. The cobordism (W; M, M′) is the trace of n-surgeries
$Display mathematics$
with null-homotopies in X representing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators of K n(M). The kernel modules K *(M), K *(M′) fit into exact sequences:
$Display mathematics$

Proposition 12.12 Let (f, b): M 2n+1X be an n-connected (2n + 1)-dimensional normal map.

1. (i) There exist presentations (e, a): (W; M, M′) → X × (I; {0}, {1}).

2. (ii) A presentation (e, a) determines a kernel (−1)n-quadratic formation for (f, b)

$Display mathematics$
The inclusion of the lagrangian,
$Display mathematics$
has components
$Display mathematics$

3. (iii) For n ⩾ 2 every formation in the stable isomorphism class is realized by a presentation of (f, b).

Proof (i) A Heegaard splitting,

$Display mathematics$
(p.313) determines k n-surgeries
The trace of the surgeries is a presentation
$Display mathematics$
The kernel formation of (f, b) given by 12.7 is
$Display mathematics$

(ii) Any presentation arises from a Heegaard splitting as in (i).

(iii) Combine (i), (ii) and 12.10. □

Note that turning a presentation of (f, b) around, and viewing it as a presentation of (f′, b′) gives the kernel formation $( H ( − 1 ) n ( F ) ; F ∗ , G )$ for (f′, b′) with

$Display mathematics$

Proposition 12.13 A kernel (− 1)n-quadratic formation of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is stably isomorphic to the boundary of a (−1)n+1-quadratic form if (and for n ⩾ 2 only if) (f, b) is bordant to a homotopy equivalence.

Proof Given a normal bordism

$Display mathematics$
with (f′, b′): M′X a homotopy equivalence make (e, a) (n + 1)-connected by surgery below the middle dimension, with kernel (−1)n+1-quadratic form (p.314) (K n+1(W), λ w, μ w). This defines a presentation of (f, b) with
$Display mathematics$
an isomorphism which is used to identify
$Display mathematics$
and
$Display mathematics$
so that the kernel formation of (f, b) is the boundary
$Display mathematics$

Conversely, suppose that n ⩾ 2 and that (f, b): M 2n+1X has a kernel formation which is stably isomorphic to the boundary of a (−1)n+1-quadratic form. By 12.12 it is possible to realize this boundary by a presentation

$Display mathematics$
with (f′, b′): M′X a homotopy equivalence. □

Example 12.14 An element

$Display mathematics$
classifies an oriented (n + 1)-plane bundle ω: S n+1BSO(n+1) with a stable trivialization δω: ω ≃ {*}: S n+1BSO. As in 5.68 use (δω,ω) to define an n-surgery (g δω, gω) on the identity degree 1 normal map 1: S 2n+1S 2n+1, with trace
$Display mathematics$
The n-connected (2n + 1)-dimensional degree 1 normal map (f, b): S(ω) → S 2n+1 has kernel (−1)n-quadratic formation over Z
$Display mathematics$
with χ(ω)=(1+(−1)(n+1)χ(δω). □

Proposition 12.15 Let

$Display mathematics$
be an (n+l)-connected (2n+2)-dimensional normal bordism between n-connected (2n + 1)-dimensional degree 1 normal maps (f, b), (f′, b′). The kernel formations (K, λ, μ; F, G), (K′, λ′, μ′; F′, G′) of (f, b), (f′, b′) are related by a stable isomorphism
$Display mathematics$

(p.315) Proof The disjoint union

$Display mathematics$
is a degree 1 normal map, where – reverses orientations. The stable isomorphism of formations is determined by the degree 1 normal map of pairs
$Display mathematics$
working as in the proof of 12.13. □

The kernel formation is also defined for an n-connected (2n+ 1)-dimensional degree 1 normal map

$Display mathematics$
which is a homotopy equivalence on the boundary, realizing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators of K n(M) as in 12.6 by a decomposition
$Display mathematics$
with
$Display mathematics$

There is also a version for normal bordisms, as follows.

Definition 12.16 Suppose given an n-connected (2n+1)-dimensional degree 1 normal bordism

$Display mathematics$
such that f| = 1: NN and f|: N′N is a homotopy equivalence. The Umkehr maps in this case are just
$Display mathematics$
the kernel Z[π1(N)]-modules are such that
$Display mathematics$
and the cobordism (M; N, N′) has an (n, n + 1)-index handle decomposition (8.23)
$Display mathematics$
(p.316)
1. (i) A Heegaard splitting for (f, b) is an expression as a union

$Display mathematics$
determined by a choice of handle decomposition as above, with
$Display mathematics$
The ith handle represents x iK n(M) = H n(M̃, Ñ), with {x 1, x 2,…, x k} ⊂ K n(M) a set of Z[π1(N)]-module generators.

2. (ii) The kernel formation associated to a Heegaard splitting as in (i) is

$Display mathematics$
with $( K n ( ∂ U ) , λ , μ ) = H ( − 1 ) n ( K n + 1 ( U , ∂ U ) ) .$  □

As in the closed case (12.6) every finite set of generators of K n(M) is realized by a Heegaard splitting of (f, b), and so determines a kernel formation (K, λ, μ; F, G).

Proposition 12.17 (Realization of formations) Let N 2n be a 2n-dimensional manifold with fundamental group π 1(N) = π, with n ⩾ 2. Every (−1)n-quadratic formation (K, λ, μ; F, G) over Z[π] is realized as a kernel formation (12.16) of an n-connected (2n + 1)-dimensional degree 1 normal bordism

$Display mathematics$
with (f, b)| = 1: NN and (f, b)|: N′N a homotopy equivalence.

(p.317) Proof By 11.51 the form (K, λ, μ) is isomorphic to the hyperbolic form $H ( − 1 ) n ( F ) ,$ so there is no loss of generality in taking $( K , λ , μ ) = H ( − 1 ) n ( F ) .$ . Let k ⩾ 0 be the rank of the f.g. free Z[π]-modules F, G, so that

$Display mathematics$
Let
$Display mathematics$
be the (2n + 1)-dimensional normal bordism defined by the trace of k trivial (n − 1)-surgeries on (f, b), with
$Display mathematics$
Realize the lagrangian of the kernel form
$Display mathematics$
by a (2n + 1)-dimensional normal bordism
$Display mathematics$
defined by the trace of k n-surgeries on (f′, b′)|, with
$Display mathematics$
The effect is a homotopy equivalence (f′, b′)|: N′N, since the kernel form is
$Display mathematics$
The n-connected (2n + 1)-dimensional degree 1 normal map of pairs
$Display mathematics$
realizes the formation (K, λ, μ; F, G). □

# 12.3 The odd-dimensional L-groups

The odd-dimensional surgery obstruction groups L 2*+1 (A) are now defined using formations. The surgery obstruction of an odd-dimensional normal map will be defined in Section 12.4 using the kernel formation of Section 12.2, and it will be proved that for n ⩾ 2 an n-connected (2n + 1)-dimensional degree 1 normal map (p.318) (f, b): M 2n+1X is bordant to a homotopy equivalence if and only if the stable isomorphism class of kernel (−1)n-quadratic formations contains the ‘boundary’ of a (−1)n+1-quadratic form (= the kernel form of the (2n + 2)-dimensional trace), in the following sense:

Definition 12.18 Let (K, λ, μ) be a (−ε)-quadratic form.

1. (i) The graph lagrangian of (K, λ, μ) is the lagrangian

$Display mathematics$
in the hyperbolic ε-quadratic form H (K)

2. (ii) The boundary of (K, λ, μ) is the graph ε-quadratic formation

$Display mathematics$
□

The graph lagrangian Γ(K, λ) and the boundary formation (K, λ, μ) depend only on the even ε-symmetric form (K, λ), and not on the ε-quadratic function μ. Note that the form (K, λ, μ) may be singular, that is the A-module morphism λ: KK * need not be an isomorphism.

Proposition 12.19

1. (i) The graphs Γ(K, λ) of (−ε)-quadratic forms (K, λ, μ) are precisely the lagrangians of H ε(K) which are the direct complements of K *.

2. (ii) An ε-quadratic formation (K, λ, μ; F, G) is isomorphic to a boundary if and only if (K, λ, μ) has a lagrangian H which is a direct complement of both the lagrangians F, G.

Proof (i) The direct complements of K * in KK * are the graphs

$Display mathematics$
of A-module morphisms λ: KK *, with
$Display mathematics$
Thus L = L if and only if λ = −∈λ *, with $μ H ∈ ( K ) ( L ) = 0$ if and only if λ admits a (−∈)-quadratic refinement μ.

(ii) For the boundary (F, φ, Θ) of a (−ε)-quadratic form (F, φ, Θ) the lagrangian F * of H ε(F) is a direct complement of both the lagrangians F, Γ(F, φ). Conversely, suppose that (K, λ, μ; F, G) is such that there exists a lagrangian H in (K, λ, μ) which is a direct complement to both F and G. By the proof of Proposition 12.5 (i) there exists an isomorphism of formations

$Display mathematics$
which is the identity on F. Now f −1 (G) is a lagrangian of H ε(F) which is a direct complement of F *, so that by (i) it is the graph Γ(F, φ) of a (−ε)-quadratic (p.319) form (F, Φ, Θ), and f defines an isomorphism of ε-quadratic formations
$Display mathematics$
□

Definition 12.20 The ε-quadratic formations (K, λ, μ; F, G), (K′, λ′, μ′; F′, G′) over A are cobordant,

$Display mathematics$
if there exists a stable isomorphism
$Display mathematics$
with B, B′ boundaries. □

Proposition 12.21

1. (i) Cobordism is an equivalence relation on ε-quadratic formations over A.

2. (ii) For any lagrangians F, G, H in a nonsingular ε-quadratic form (K, λ, μ)

$Display mathematics$

3. (iii) For any ε-quadratic formation (K, λ, μ; F, G)

$Display mathematics$

Proof (i) Clear.

(ii) (Taken from Proposition 9.14 of [69]). Choose lagrangians F *, G *, H * in (K, λ, μ) complementary to F, G, H, respectively. The ε-quadratic formations (K i, λ i, μ i; F i, G i) (1 ≤ i ≤ 4) defined by

$Display mathematics$
are such that
$Display mathematics$
Each of (K i, λ i, μ i; F i, G i) (1 ≤ i ≤ 4) is isomorphic to a boundary, since there exists a lagrangian H i in (K i, λ i, μ i) complementary to both F i and G i, so that (p.320) 12.19 (ii) applies and (K i, λ i, μ i; F i, G i) ˜ 0. Explicitly, take
$Display mathematics$

(iii) By (ii)

$Display mathematics$
and by Proposition 12.5 (iii)
$Display mathematics$
□

Remark 12.22 The identity of 12.21 (ii)

$Display mathematics$
is the L-theoretic analogue of the Whitehead Lemma 8.2. See Lemma 6.2 of Wall [92] and the commentary on pp. 72–73 of [92] for the geometric motivation. □

Definition 12.23 The (2n + 1)-dimensional L-group L 2n+1(A) of a ring with involution A is the group of cobordism classes of (−1)n-quadratic formations (K, λ, μ; F, G) over A, with addition and inverses given by

$Display mathematics$
□

Since L 2n+1(A) depends on the residue n(mod 2), only two L-groups have actually been defined, L 1(A) and L 3(A).

Example 12.24 Kervaire and Milnor [38] proved that the odd-dimensional L-groups of Z are trivial

$Display mathematics$
See Example 12.44 below for an outline of the computation. □

Remark 12.25 Chapter 22 of Ranicki [71] is an introduction to the computation of the odd-dimensional surgery obstruction groups of finite groups π, with

$Display mathematics$
See Hambleton and Taylor [30] for a considerably more complete account. □

(p.321) Example 12.26 The odd-dimensional L-groups of Z[Z 2] with the oriented involution = T are given by

$Display mathematics$
□

# 12.4 The odd-dimensional surgery obstruction

It was shown in Section 10.4 that every (2n+1)-dimensional degree 1 normal map is bordant to an n-connected degree 1 normal map. As in the even-dimensional case considered in Section 11.5 there is an obstruction to the existence of a further bordism to an (n + 1)-connected degree 1 normal map (= homotopy equivalence), which is defined as follows.

Definition 12.27 The surgery obstruction of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is the cobordism class of a kernel (−1)n-quadratic formation over Z[π1(X)]:

$Display mathematics$
□

The main result of this section is that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is bordant to a homotopy equivalence. It is clear that if (f, b) is a homotopy equivalence then σ * (f, b) = 0, for then (K, λ, μ; F, G) is a trivial formation.

Proposition 12.28 The surgery obstructions of bordant n-connected (2n + 1)-dimensional degree 1 normal maps (f, b): M 2n+1X, (f′, b′): M′ 2n+1X are the same:

$Display mathematics$

Proof By 12.15 the kernel formations (K, λ μ; F, G), (K′, λ′, μ′; F′, G′) of (f, b), (f′, b′) are related by a stable isomorphism

$Display mathematics$
with (K n+1(W), λ W, μ W) the kernel (−1)n+1-quadratic form of an (n + 1)-connected normal bordism
$Display mathematics$
By Proposition 12.21 (iii)
$Display mathematics$
(This can also be proved geometrically, by considering the (n + 1)-connected normal bordism (e, a) obtained from
$Display mathematics$
(p.322) by n-surgeries on the interior killing K n(M × I) = K n(M), with a stable isomorphism
$Display mathematics$
as above.) The surgery obstructions are such that
$Display mathematics$
□

Theorem 12.29 A (2n + 1)-dimensional degree 1 normal map of pairs

$Display mathematics$
with ∂f: ∂M∂X a homotopy equivalence has a rel ∂ surgery obstruction
$Display mathematics$
such that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is bordant rel ∂ to a homotopy equivalence of pairs.

Proof The surgery obstruction of (f, b) is defined by

$Display mathematics$
with (K′, λ′, μ′; F′, G′) a kernel (−1)n-quadratic formation for any n-connected degree 1 normal map (f′, b′): (M′, ∂M) → (X, ∂X) bordant to (f, b) relative to the boundary, with ∂f′ = ∂f, exactly as in the closed case ∂M = ∂ X = ∅ in 12.27. The rel version of 12.28 shows that the surgery obstruction is a normal bordism invariant, which is 0 for a homotopy equivalence. Conversely, assume that n ⩾ 2 and σ *(f, b) = 0 ∈ L 2n+1(Z[π1(X)]), so that (f, b) has a kernel (−1)n-quadratic formation (K, λ, μ; F, G) with a stable isomorphism,
$Display mathematics$
for some boundary formations B = (H, Φ, Θ), B′ = (H′, Φ′, Θ′), with H, H′ f.g. free Z[π1(X)]-modules of ranks k, k′ (say). As in the proof of Proposition 11.42 use the (−1)n+1-quadratic form (H, Φ, Θ) to perform k n-surgeries on (f, b): MX killing 0 ∈ K n(M), such that the trace,
$Display mathematics$
is (n + 1)-connected with kernel (−1)n+1-quadratic form
$Display mathematics$
The effect is an n-connected degree 1 normal map (f′, b′): M′X with kernel formation (K, λ, μ; F, G) ⊕ B stably isomorphic to the boundary B′. By Proposition 12.13 K n(M′) can be killed by k′ n-surgeries on (f′, b′), so that (f, b) is bordant to a homotopy equivalence. □

(p.323) Corollary 12.30 Let π be a finitely presented group with an orientation character w: π → Z2, and let n ⩾ 2. Every element xL 2n+1(Z[π]) is the rel ∂ surgery obstruction x = σ *(f, b) of an n-connected (2n + 1)-dimensional degree 1 normal bordism (f, b): M 2n+1X with1(X), w(X)) = (π, w).

Proof By Proposition 11.75 there exists a closed 2n-dimensional manifold N with (π1(N), w(N)) = (π, w). By Proposition 12.17 every (−1)n-quadratic formation (K, λ, μ; F, G) representing x is realized as the kernel formation of an n-connected (2n + 1)-dimensional degree 1 normal bordism,

$Display mathematics$
with (f, b)| = 1: N → N and (f, b)|: N′ → N a homotopy equivalence. The rel surgery obstruction is
$Display mathematics$
□

Remark 12.31 An ε-quadratic formation over a ring with involution A is null-cobordant if and only if it is stably isomorphic to the boundary of a (−∈)- quadratic form (Corollary 9.12 of Ranicki [75]). A (−l)n-quadratic formation (K, λ, μ; F, G) is, thus, such that

$Display mathematics$
if and only if (K, λ, μ; F, G) is stably isomorphic to the boundary (H, Φ, Θ) of a (−1)n+1-quadratic form (H, Φ, Θ). For a group ring A = Z[π] this can be proved geometrically, using Theorem 12.29 and Corollary 12.30. □

# 12.5 Surgery on formations

The odd-dimensional surgery obstruction theory developed in Section 12.4 is somewhat indirect—it is hard to follow through the algebraic effect of geometric surgeries. This will now be made easier, using algebraic surgery on formations.

If (f, b): M → X, (f′, b′): M′ → X are n-connected (2n + 1)-dimensional degree 1 normal maps such that (f′, b′) is obtained from (f, b) by an n-surgery then a kernel (−l)n-quadratic formation for (f′, b′) can be obtained by an algebraic surgery on a kernel (−1)n-quadratic formation for (f, b). The geometric surgeries on (f, b) correspond to algebraic surgeries on a kernel formation, as in the even-dimensional case considered in Section 11.3. However, odd-dimensional surgery behaves somewhat differently from even-dimensional surgery. In both cases, the aim of performing surgery is to make the kernel modules as small as possible. Given an ε-quadratic form (K, λ, μ) over A it is possible to kill an element xK if and only if μ(x) = 0, with unique effect: if x ≠ 0 generates a (p.324) direct summand <x> ⊂ K the effect of the surgery is a cobordant form (K′, λ′, μ′) with

$Display mathematics$

Given an ε-quadratic formation (K, λ, μ; F, G) it is possible to kill every element in the kernel module xK/(F + G) by algebraic surgery, but there are many choices in carrying out such a surgery, and the effect of any such surgery may result in a formation (K′, λ′, μ′; F′, G′) with kernel module K′/(F′ + G′) bigger than K/(F+G). In the context of geometric surgery consider the trace of k n-surgeries on an n-connected (2n+1)-dimensional normal map (f, b): MX killing x 1, x 2,…,x kK n(M):

$Display mathematics$
and let (K,λ, μ; F, G), (K′, λ′, μ′; F′, G′) be kernel (−1)n-quadratic formations for (f, b), (f′, b′). Proposition 10.25 (iii) gives a commutative braid of exact sequences
and a set of Z[π1(X)]-module generators {x′ 1, x′ 2,…,x′ k} ⊂ ker(K n(M′) → K n(W)) with
$Display mathematics$
The different effects of killing x 1, x 2,…,x k correspond to the different ways of framing n-embeddings g i: S nM 2n+1 representing x i, or equivalently to (p.325) the different extensions of g i to framed n-embeddings i: S n × D n+1M 2n+1. Every set of Z[π1(X)]-module generators {x 1,x 2,…,x k} ⊂ K n(M) can be killed by n-surgeries with (n + 1)-connected trace (i.e. K n(W) = 0) but in general the effect (f′, b′): M′ → X will not be a homotopy equivalence, with K n(M′) ≠ 0.

In order to keep track of algebraic surgeries on formations it is convenient to work with the following refinement of the notion of a formation.

Definition 12.32 (i) A split ε-quadratic formation over A

$Display mathematics$
is given by f.g. free A-modules F, G, morphisms γ: GF, δ: GF * and a (−∈)-quadratic form (G, Θ) such that
1. (a) γ * δ = Θ − ∈Θ*: GG *,

2. (b) the sequence

$Display mathematics$
is exact.

Equivalently,
$Display mathematics$
is a morphism of split ε-quadratic forms which is the inclusion of a lagrangian.

(ii) An isomorphism of split ε-quadratic formations over A

$Display mathematics$
is given by isomorphisms α: FF′, β: GG′ and a (−∈)-quadratic form (F *, χ) such that the diagram
commutes. Thus,
$Display mathematics$
is an isomorphism of hyperbolic ε-quadratic forms with f(F) = F′, f(G) = G′.

(p.326) (iii) A split ε-quadratic formation (F, G) is trivial if it is isomorphic to

$Display mathematics$

(iv) A stable isomorphism of split ε-quadratic formations over A

$Display mathematics$
is an isomorphism of the type
$Display mathematics$
with (H, H *), (H′, H′ *) trivial split formations.

(v) The boundary of a split (−∈)-quadratic form (K, ψ) is the graph split ε-quadratic formation

$Display mathematics$

(vi) Split ε-quadratic formations (F,G), (F′,G′) are cobordant if there exists a stable isomorphism

$Display mathematics$
for some split (−∈)-quadratic forms (K, ψ), (K′,ψ′). □

Proposition 12.33

1. (i) A split ε-quadratic formation (F, G) is isomorphic to a boundary if and only if there exists a split (−∈)-quadratic form (F *, χ) such that the morphism γ + (χ − εχ*)δ: GF is an isomorphism.

2. (ii) A split formation (F, G) is stably isomorphic to 0 if and only if δ: G → F* is an isomorphism.

3. (iii) Cobordism is an equivalence relation on split ε-quadratic formations over A.

4. (iv) The cobordism group of split (−1)n-quadratic formations over A is isomorphic to L 2n+1(A).

5. (v) For any split (−1)n-quadratic formation $( F , G ) = ( F , ( ( γ δ ) , θ ) G )$ over A

$Display mathematics$

(p.327) Proof (i) If (α, β, χ): (F, G) → ∂(K,ψ) is an isomorphism of split formations then

$Display mathematics$
is an A-module isomorphism.

For the converse, consider first the special case when γ is an isomorphism. There is defined an isomorphism of split formations

$Display mathematics$
More generally, if γ′ = γ+(χ−∈χ *)δ: GF is an isomorphism there is defined an isomorphism of split formations
$Display mathematics$
and (F′, G′) is isomorphic to a boundary by the special case.

(ii) If (α, β, X): (F, G) → (K, K *) is an isomorphism of split formations then

$Display mathematics$
is an A-module isomorphism.

Conversely, if δ is an isomorphism there is defined an isomorphism of split formations,

$Display mathematics$

(iii) Clear.

$Display mathematics$
and an isomorphism of split formations determines an isomorphism of formations. Every ε-quadratic formation (K, λ, μ; F, G) is isomorphic to one of this type, by 12.3, and a (stable) isomorphism of formations
$Display mathematics$
lifts to a (stable) isomorphism of split formations
$Display mathematics$
So the only essential difference between a formation and a split formation is the choice of ‘Hessian’ form Θ. Suppose, given split ε-quadratic formations
$Display mathematics$
(p.328) with different Θ, Θ′ such that
$Display mathematics$
Let
$Display mathematics$
be the split ε-quadratic formation given by an extension (provided by 11.51) of the inclusion of the lagrangian,
$Display mathematics$
to an isomorphism of hyperbolic split ε-quadratic forms
$Display mathematics$
$Display mathematics$
is isomorphic to a boundary by (i), since
$Display mathematics$
is an isomorphism. Similarly for the split formation Φ′ ⊕ Φ˜. The split formation Φ ⊕ Φ′ ⊕ Φ˜ is cobordant to both Φ and Φ′, which are thus cobordant to each other.

(v) These identities follow from 12.21 (ii)+(iii). (Alternatively, note that there exist stable isomorphisms

$Display mathematics$
The construction of such stable isomorphisms are exercises for the reader.) □

Definition 12.34 The data (H, X, j) for an algebraic surgery on a split ε-quadratic formation (F, G) is a split (−∈)-quadratic form (H, X) together with a (p.329) morphism j: F → H *. The effect of the algebraic surgery is the split ε-quadratic formation (F′, G′) with

$Display mathematics$
□

Proposition 12.35 (i) If (F 1, G 1), (F 2, G 2), (F 3, G 3) are split ε-quadratic formations such that (F i+1, G i+1) is stably isomorphic to the effect of an algebraic surgery on (F i, G i) (i = 1, 2) then (F 3, G 3) is stably isomorphic to the effect of an algebraic surgery on (F 1, G 1).

(ii) Split ε-quadratic formations (F, G), (F′, G′) are cobordant if and only if (F′, G′) is stably isomorphic to the effect of an algebraic surgery on (F, G).

(iii) A split (−1)n-quadratic formation (F, G) is such that (F, G) = 0 ∈ L 2n+1(A) if and only if there exist algebraic surgery data (H, χ, j) such that

$Display mathematics$
is an isomorphism, in which case (F, G) is stably isomorphic to the boundary $∂ ( G ⊕ H , ( θ 0 j γ χ ) ) .$

Proof (i) Exercise for the reader!

(ii) Suppose first that (F, G), (F′, G′) are cobordant, so that there exists a stable isomorphism

$Display mathematics$
for some (—∈)-quadratic forms (H, χ), (H′, χ′). Now (F, G)⊕∂(H, χ) is the effect of the algebraic surgery on (F, G) with data (H, χ, 0), and (F′, G′) is stably isomorphic to the effect of the algebraic surgery on (F′, G′) ⊕ ∂ (H′, χ′) with data (H′, χ′, j′ = (0 1): F′H′ *H′ *). It now follows from (i) that (F′, G′) is stably isomorphic to the effect of an algebraic surgery on (F, G).

Conversely, suppose that (F′, G′) is the effect of an algebraic surgery on (F, G) with data (H, χ, j). By 12.33 (v) (F′, G′) is cobordant to the split formation

$Display mathematics$
Now (F′ *, G′) is isomorphic to (F *, G) ⊕ (H *, H), and (F *, G) is cobordant to (F, G) (by 12.33 (v)), so that (F′, G′) is cobordant to (F, G).

(p.330) (iii) By (ii) a split (−1)n-quadratic formation (F, G) is null-cobordant if and only if there exists data (H, χ, j) such that the effect of the algebraic surgery (F′, G′) is trivial. □

All this can now be applied to surgery on highly-connected odd-dimensional normal maps.

Proposition 12.36 Let (f, b): M 2n+1X be an n-connected (2n+1)-dimensional degree 1 normal map. (i) A presentation (12.11) of (f, b),

$Display mathematics$
determines a kernel split (−1)n -quadratic formation over Z[π1(X)]
$Display mathematics$
with
$Display mathematics$
and exact sequences
$Display mathematics$

(ii) The surgery obstruction of (f, b) is the cobordism class

$Display mathematics$
of the kernel split formation (F, G) constructed in (i) from any presentation of (f, b).

(iii) The effect of l simultaneous geometric n-surgeries on (f, b) killing x 1, x 2, …, x lK n(M) is a bordant n-connected (2n + 1)-dimensional normal map (f′, b′): M′ → X with kernel split formation (F′, G′) obtained by algebraic surgery on (F, G) with data (H, χ, j) such that

$Display mathematics$

(iv) For n ⩾ 2 algebraic surgeries on (F, G) are realized by geometric surgeries on (f, b).

(p.331) (v) Let x 1, x 2, …, x 1K n+1(M) be as in (iii) (or (iv)). If there exist y 1, y 2, …, y lK n+1(M) such that

$Display mathematics$
with λ: K n(M) × K n+1(M) → Z[π1(X)] the homology intersection pairing (10.22) then
$Display mathematics$

Proof (i) The split formation (F, G) is just the split version of the kernel formation 12.12 (ii).

(ii) Immediate from (i) and 12.29.

(iii) Let (f′, b′): M′ 2n+1X be the effect of l n-surgeries on (f, b) killing x 1, x 2, …, x lK n(M). The trace degree 1 normal bordism,

$Display mathematics$
is n-connected, and such that
$Display mathematics$
Given a presentation (e, a) of (f, b) as in (i) define a presentation of (f′, b′)
$Display mathematics$
The corresponding kernel split (−1)n-quadratic formation (F′, G′) for (f′, b′) is the effect of an algebraic surgery on the kernel (F, G) in (i) with data (H, χ, j: F → H *) such that
$Display mathematics$
(p.332) with exact sequences of Z[π1(X)]-modules,
$Display mathematics$
and a commutative braid of exact sequences:
The normal map (e, a): NX × I is (n + 1)-connected if and only if the Z[π1(X)]-module morphism,
$Display mathematics$
is onto, in which case K n+1(N) is a stably f.g. free Z[π1(X)]-module and the kernel (−1)n+1-quadratic form is given by
$Display mathematics$

(iv) Suppose given a presentation (e, a) of (f, b) as in (i) and data (H, χ, j) for algebraic surgery on the kernel split (−1)n-quadratic formation (F, G) with (p.333) H = Z[π1(X)]l and effect (F′, G′). Let x 1,x 2,…,x lK n(M) be the images of the basis elements (0,…, 0, 1, 0, …, 0) ∈ H under the composite

$Display mathematics$
Use the b-framing section $s b f r : I n + 1 ( f ) → I n + 1 f r ( f )$ (10.14) to identify each
$Display mathematics$
with a regular homotopy class of framed n-immersions in (f, b). As in the proof of Proposition 10.25, x i contains framed n-embeddings. For n ⩾ 2 the framed n-embeddings can be varied within the regular homotopy class by arbitrary elements of $Q ( − 1 ) n + 1 ( ℤ [ π 1 ( X ) ] ) ,$ as in the proof of Proposition 11.42. It is, therefore, possible to kill x 1, x 2, …, x lK n(M) by n-surgeries on (f, b) such that (F′, G′) is the kernel split (−1)n-quadratic formation for (f′, b′) obtained as in (iii).

(v) The Z[π1(X)]-module morphism

$Display mathematics$
is onto, so that the braid in (iii) identifies K n(M′) with the cokernel of the Z[π1(X)]-module morphism
$Display mathematics$
□

Remark 12.37 For any normal bordism between n-connected (2n+1)-dimensional normal maps (f, b), (f′, b′),

$Display mathematics$
it is possible to kill the kernel Z[π1(X)]-modules K i(N) (in) by surgery below the middle dimension. Thus (g, c): NX × I can be made (n + 1)-connected, with K n+1(N, M) a f.g. free Z[π1(X)]-module of rank l ⩾ 0 (say). The cobordism (N; M, M′) is the trace of l n-surgeries on (f, b) with geometric effect (f′, b′), and with algebraic effect given by 12.36. □

Algebraic surgery on formations can also be used for purely algebraic computations:

Proposition 12.38 Let A be a principal ideal domain with involution, with quotient field K. (p.334)

1. (i) Every split ε-quadratic formation (F, G) over A is cobordant to a formation (F′,G′) with δ′: G′ → F′ * injective.

2. (ii) L 2n+1(K) = 0.

Proof (i) The A-module coker(δ: GF *) is finitely generated, so that it can be expressed as a direct sum of a f.g. free A-module S and a f.g. torsion A-module T:

$Display mathematics$
(Here, torsion means that aT = 0 for some a ≠ 0 ∈ A.) The sesquilinear pairing,
$Display mathematics$
is such that for any x ∈ S, there exists y ∈ ker(δ) such that λ(x,y) = 1 ∈ A. The abstract version of 12.36 (iv) shows that for any algebraic surgery on (F,G) with data (H, X, j) such that
$Display mathematics$
the effect of the algebraic surgery (F′,G′) has δ′: G′ → F′ * injective with
$Display mathematics$

(ii) Take A = K, ∈ = (−1)n in (i). The only torsion A-module is 0, so δ′ is an isomorphism, and (F′, G′) is a trivial split (−1)n-quadratic formation. □

Odd-dimensional surgery obstructions were originally formulated by Kervaire and Milnor [38] and Wall [90] in terms of linking forms, but the method only applies to finite fundamental groups π. However, linking forms remain useful tools in surgery theory. This section is a brief introduction to linking forms and their use in the computation L 2*+1(Z) = 0. See Chapter 3 of Ranicki [70] for a considerably more complete account.

Let A be a ring with involution, and let SA be a subset such that

1. (i) each sS is a central non-zero divisor, with S,

2. (ii) if s, tS then stS,

3. (iii) 1 ∈ S.

Definition 12.39 (i) The localization of A is the ring of fractions

$Display mathematics$
with elements denoted $α s .$ The natural map
$Display mathematics$
is an injective morphism of rings with involution.

(p.335) (ii) An (A, S)-module T is a f.g. A-module such that sT = 0 for some sS, and which admits a f.g. free A-module resolution of the type

$Display mathematics$
The dual of T = coker(d) is the (A, S)-module
$Display mathematics$

(iii) A nonsingular ε-symmetric linking form (T, λ) over (A, S) is an (A, S)- module T together with a sesquilinear pairing

$Display mathematics$
such that for all x, y, zT; a, bA
1. (a) λ(x, y + z) = λ(x, y) + λ(x, z) ∈ S −1 A/A,

2. (b) λ(ax, by) = (x, y)āS −1 A/A,

3. (c) $λ ( y , x ) = ∈ λ ( x , y ) ¯ ∈ S − 1 A / A ,$

4. (d) the adjoint A-module morphism

$Display mathematics$
is an A-module isomorphism.

(iv) A nonsingular split ε-quadratic linking form (T, λ, v) over (A, S) is an ε-quadratic linking form (T, λ) together with a function

$Display mathematics$
such that
1. (a) ν(ax)=ā∈Q (S −1 A/A)(x∈T,a∈A),

2. (b) ν(x+y)−ν(x)−ν(y)=λ(x,y)∈Q (S −1 A/A)(x,y∈T)

3. (c) $λ ( x , x ) = v ( x ) + ∈ v ( x ) ¯ ∈ S − 1 A A ( x ∈ T ) .$

□

Example 12.40 Let A be an integral domain, and let S = A\{0} ⊂ A.

1. (i) The localization S −1 A = K is the quotient field of A.

2. (ii) If A is a principal ideal domain an (A, S)-module T is a f.g. A-module such that sT = 0 for some sS. For any f.g. A-module H the torsion submodule THH is an (A, S)-module, and H/TH is a f.g. free A-module. □

There is a close connection between formations and linking forms:

Proposition 12.41 The isomorphism classes of nonsingular split ε-quadratic linking forms (T, λ, v) over (A, S) are in one–one correspondence with the stable (p.336) isomorphism classes of split (−∈)-quadratic formations $( F , ( ( γ δ ) , θ ) G )$ over A such that δ: S –1 GS −1 F * is an S −1 A-module isomorphism.

Proof Given such a split formation (F, G) define a linking form (T, λ, v) by

$Display mathematics$